Please excuse my lack of ITIL knowledge as I ask this question.
I am hoping someone can point me to some concrete best practice documentation that supports my argument and provides benefits to assigning an incident to yourself once you initiate the first contact with the user.

Currently, my team has a team "bucket" with incidents assigned to the team but no individual person on that team and we each have our own personal "buckets" with incidents assigned to use.

The idea is well all address issues in the team bucket and communicate with the user but never really ever take ownership of an incident we are communicating on. This leads to incidents getting buried and getting lost in the shuffle.

I am proposing that once the first contact is made, we assign the incident to ourselves to take ownership and see the incident through to the resolution.

I guess I should ask if this is even best practice and if so, where I can find that documented somewhere so I can use it in my proposal?

I really appreciate your help!

I would be what you'd probably call level 2 support where my team specializes in one specific application in our application suite.